Localized waves in nonlinear oscillator chains

Iooss, Gérard; James, Guillaume

doi:10.1063/1.1836151

Cited by 62 publications

(106 citation statements)

References 63 publications

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“…Being an obstacle to the "translationally invariant" kinks, the Peierls-Nabarro barrier is also detrimental to the existence of sliding kinks -at least for small c (see reviews in [S03] and [IJ05]). …”

Section: Introductionmentioning

confidence: 99%

Travelling kinks in discrete phi⁴models

2005

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Abstract. In recent years, three exceptional discretizations of the φ 4 theory have been discovered [SW94, BT97, K03] which support translationally invariant kinks, i.e. families of stationary kinks centred at arbitrary points between the lattice sites. It has been suggested that the translationally invariant stationary kinks may persist as sliding kinks, i.e. discrete kinks travelling at nonzero velocities without experiencing any radiation damping. The purpose of this study is to check whether this is indeed the case. By computing the Stokes constants in beyond-all-order asymptotic expansions, we prove that the three exceptional discretizations do not support sliding kinks for most values of the velocity -just like the standard, one-site, discretization. There are, however, isolated values of velocity for which radiationless kink propagation becomes possible. There is one such value for the discretization of [SW94] and three sliding velocities for the model of [K03].

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Section: Introductionmentioning

confidence: 99%

Travelling kinks in discrete phi⁴models

2005

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“…[8,9]), chains of nonlinear oscillators (see e.g. [10]) with randomly distributed frequencies, and models of quantum chaos with a kicked soliton [11] and a kicked BEC [12,13,14].…”

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confidence: 99%

Destruction of Anderson Localization by a Weak Nonlinearity

2008

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We study numerically a spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schrödinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time ∝ t α , with the exponent α being in the range 0.3 − 0.4. For small nonlinearities the distribution remains localized in a way similar to the linear case. [2,3,4,5,6]. An interesting new aspect in such systems is an importance of nonlinear effects since in a good approximation the evolution of BEC can be described by the nonlinear Gross-Pitaevskii equation (see e.g. [7]). An interplay of disorder, localization, and nonlinearity appears also in other physical systems like wave propagation in nonlinear disordered media (see e.g. [8,9]), chains of nonlinear oscillators (see e.g.[10]) with randomly distributed frequencies, and models of quantum chaos with a kicked soliton [11] and a kicked BEC [12,13,14].We focus here on the discrete Anderson nonlinear Schrödinger equation (DANSE)where β characterizes nonlinearity, V is hopping matrix element, on-site energies are randomly distributed in the range −W/2 < E n < W/2, and the total probability is normalized to unity n | ψ n | 2 = 1. For β = 0 and weak disorder all eigenstates are exponentially localized with the localization length l ≈ 96(V /W ) 2 at the center of the energy band [19]. Hereafter we set for convenience = V = 1, thus the energy coincides with the frequency. For nonlinear equation (1) we consider the following problem: how an initially localized field | ψ n (0) | 2 = δ n,0 is spreading? In the linear case the spreading saturates after excitation of all linear modes that have significant values at n = 0. The same process of "initial excitation" of modes happens in the nonlinear case as well, this initial stage of spreading has been analyzed recently in refs. [20,21] and is now well understood. However, a behavior at large time scales remains less clear. The results presented in [20] support the view of eventual exponential localization of the field. We demonstrate below that the spreading is unlimited, however it is rather slow -subdiffusive.The basic idea is to use the equivalence between the Anderson localization and the localization of quasienergy eigenstates in a kicked quantum rotator [15,16]. In the latter model the case of quantum chaos with nonlinearity has been considered analytically and numerically in [17,18] and it has been shown that above a certain nonlinearity level, nonlinear phase shifts lead to a complete delocalization with a subdiffusive spreading over all states [17]. Furthermore it has been argued that the same situation should appear for the DANSE (1).We first apply the theoretical arguments of paper [17] to model (1), and then perform large scale numerical simulations of a wave packet spreading on a time scale which is by 5-6 orders of magnitude larger compared to that i...

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“…An abundance of works on such solutions in sole nonlinear chains (without coupling the main system) can be found in the literature. For example, the existence of breathers has been discussed by Cretegny et al [40], Iooss and James [41], or by James et al [42] in FermiPasta-Ulam chains with Hertz contact, and more generally in nonlinear lattices by Aubry [43]. Similarly, Starosvetsky and Manevitch [44] have studied localization and energy exchange in a periodic dimer FermiPasta-Ulam chain, while Perchikov and Gendelman [45] have discussed existence and stability of discrete breathers in a chain with vibro-impact potentials.…”

Section: Discussionmentioning

confidence: 99%

Vibratory control of a linear system by addition of a chain of nonlinear oscillators

2017

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An N -degree-of-freedom model consisting of a single-degree-of-freedom linear system coupled to a chain of (N − 1) light nonlinear oscillators is studied. The connection between the chain and the singledegree-of-freedom system is supposed to be linear. Time multi-scale system behaviors at fast and slow time scales are investigated and lead to the detection of the slow invariant manifold and equilibrium and singular points. These points correspond to periodic regimes and strongly modulated responses, respectively. These analytical developments are used to provide evidence of transfer of vibratory energy of the main system to the chain in the form of localized modes during periodic regimes and extreme energy exchanges between modes when the overall structure faces singularities. Furthermore, analytical predictions at slow time scale and nonlinear normal modes of the system are compared with numerical results obtained from direct time integration of the system equations, showing a good agreement between them. Finally, we present a procedure showing how these analytical developments can be used to study a system where the main structure is replaced by a multi-degree-of-freedom linear system, by projecting its dynamics on one of its modes.

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Localized waves in nonlinear oscillator chains

Cited by 62 publications

References 63 publications

Travelling kinks in discrete phi⁴models

Travelling kinks in discrete phi⁴models

Destruction of Anderson Localization by a Weak Nonlinearity

Vibratory control of a linear system by addition of a chain of nonlinear oscillators

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Localized waves in nonlinear oscillator chains

Cited by 62 publications

References 63 publications

Travelling kinks in discrete phi4models

Travelling kinks in discrete phi4models

Destruction of Anderson Localization by a Weak Nonlinearity

Vibratory control of a linear system by addition of a chain of nonlinear oscillators

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Travelling kinks in discrete phi⁴models

Travelling kinks in discrete phi⁴models